If `log_(10 ) x - log_(10) sqrt(x) = (2)/(log_(10 x))`. The value of x is

Options

A. `10^(-3 )`
B. `10^(-2)`
C. `(1)/(10)`
D. 10

Answer

To solve the equation \( \log_{10} x - \log_{10} \sqrt{x} = \frac{2}{\log_{10} x} \), we will follow these steps:

### Step 1: Simplify the left side of the equation
We know that \( \sqrt{x} = x^{1/2} \). Therefore, we can rewrite the logarithm:
\[ \log_{10} \sqrt{x} = \log_{10} (x^{1/2}) = \frac{1}{2} \log_{10} x \]
Now substitute this back into the equation:
\[ \log_{10} x - \frac{1}{2} \log_{10} x = \frac{2}{\log_{10} x} \]

### Step 2: Combine the logarithms
Now, combine the logarithms on the left side:
\[ \frac{1}{2} \log_{10} x = \frac{2}{\log_{10} x} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ \left(\frac{1}{2} \log_{10} x\right) \cdot \log_{10} x = 2 \]
This simplifies to:
\[ \frac{1}{2} (\log_{10} x)^2 = 2 \]

### Step 4: Multiply both sides by 2
To eliminate the fraction, multiply both sides by 2:
\[ (\log_{10} x)^2 = 4 \]

### Step 5: Take the square root of both sides
Taking the square root gives us two cases:
\[ \log_{10} x = 2 \quad \text{or} \quad \log_{10} x = -2 \]

### Step 6: Solve for x in both cases
1. For \( \log_{10} x = 2 \):
\[ x = 10^2 = 100 \]
2. For \( \log_{10} x = -2 \):
\[ x = 10^{-2} = \frac{1}{100} = 0.01 \]

### Final Answer
The values of \( x \) are \( 100 \) and \( 0.01 \).

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